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Theorem equvinv 29114
Description: Similar to equvini 1927 without using ax12o 1875. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equvinv  |-  ( x  =  y  <->  E. z
( x  =  z  /\  z  =  y ) )
Distinct variable groups:    x, z    y, z

Proof of Theorem equvinv
StepHypRef Expression
1 equcomi 1646 . . . . 5  |-  ( x  =  z  ->  z  =  x )
2 equcomi 1646 . . . . 5  |-  ( z  =  x  ->  x  =  z )
31, 2impbii 180 . . . 4  |-  ( x  =  z  <->  z  =  x )
43anbi1i 676 . . 3  |-  ( ( x  =  z  /\  z  =  y )  <->  ( z  =  x  /\  z  =  y )
)
54exbii 1569 . 2  |-  ( E. z ( x  =  z  /\  z  =  y )  <->  E. z
( z  =  x  /\  z  =  y ) )
6 ax-17 1603 . . 3  |-  ( x  =  y  ->  A. z  x  =  y )
7 equequ1 1648 . . 3  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
86, 7equsexv-x12 29113 . 2  |-  ( E. z ( z  =  x  /\  z  =  y )  <->  x  =  y )
95, 8bitr2i 241 1  |-  ( x  =  y  <->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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